Method approximates planar curves with circular arcs of equal length.
problem Approximating planar curves with circular arcs of equal length.
method Proposed by I.Kh. Sabitov and A.V. Slovesnov, extended with new inequalities and computer modeling.
result Derived inequalities for the length of a convex spiral arc with prescribed Hermite data.
Projected random forests improve circular data prediction with adaptive arc length and finite-sample coverage.
problem Regression with circular responses.
method Adapting linear-response models to circular data using projection and random forest out-of-bag mechanism.
result Projected random forest out-of-bag conformal prediction sets are more efficient and shorter than alternative methods.
A new method joins two arcs with a degree of freedom.
problem Joining two arcs with a precise point.
method Geometric approach using tangent vectors and points.
result A novel method to determine the join point.
Curve diffusion flow straightens curves with endpoints on intersecting lines.
problem Straightening open curves with endpoints on intersecting lines.
method Curve diffusion flow with mixed boundary conditions.
result The curve converges to a circular arc of the same length.
Study convex embeddability in linear and circular orders, applying to knots.
problem Understanding the quasi-order of convex embeddability in linear and circular orders.
method Combinatorial and descriptive set-theoretic methods applied to arcs and knots.
result Established combinatorial properties and lower bounds for knot complexity.
A lens cluster minimizes perimeter in the plane with given area constraints.
problem Minimizing perimeter in the plane with given area constraints.
method Analyzing lens clusters consisting of circular arcs with specific geometric properties.
result Lens clusters are local minimizers of the total perimeter functional.
The paper identifies all surfaces with two transversal circular arcs through each point.
problem Finding surfaces with two transversal circular arcs through each point.
method Uses a new factorization technique for quaternionic polynomials.
result Proves all such surfaces are images of subsets of specific sets under inversions.
We describe a general family of curved-crease folding tessellations consisting of a repeating "lens" motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this init…
New insights into minimal surfaces in H^2 x R foliated by arcs and their Jacobi fields.
problem Minimal surfaces in H^2 x R foliated by arcs and their properties.
method Analysis of catenoids, parabolic catenoids, and tall rectangles; study of Jacobi operator and fields.
result Computations of Jacobi fields for deformations to other surfaces in the family.
European steel industry shifts to electric arc furnaces, reducing scrap imports and increasing competition.
problem Reducing CO2 emissions in the European steel industry through electric arc furnaces.
method Combining trade data with business intelligence to model the impact of EAF capacity on scrap trade.
result Scrap imports decrease as EAF capacity increases, highlighting the need for a new business ecosystem.
The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.
problem Finding the unique minimizer of area for hyperbolic bodies with curvature constraints.
method Introduced the concept of 'thick λ-sausage' bodies and used extra assumption of thickness to handle non-convex inner parallel bodies. result The thick λ-sausage body is the unique minimizer of area among all bodies with a given length and curvature constraints. The paper extends curve curvature types from normed to gauge planes.
problem Extending curve curvature classification to gauge planes.
method Using gauge analogue of Birkhoff orthogonality and differential geometry.
result Four curvature types (Minkowski, normal, circular, arc-length) are identified.
Study proves quantitative results for isoperimetric problem outside convex bodies in the plane.
problem Quantitative estimates for the relative isoperimetric problem outside convex bodies in the plane.
method Flow approach and Łojasiewicz estimates to prove quantitative stability for minimizers.
result Explicit constants and optimal exponents/rates for Łojasiewicz estimates and rates of convergence for gradient flow.
Study circular tractrices and pseudospheres in 3D space.
problem Geometric properties of circular tractrices and pseudospheres.
method Exploration of geometric properties through analysis.
result Characterization of circular tractrices and pseudospheres.
The paper defines circular orderability for quandles and explores their properties.
problem Understanding the structure of quandles through circular orderings.
method Introduced circular orderability for quandles, showed embedding properties, and provided examples.
result Spaces of circular orderings embed in the space of all orderings, and examples of non-circularly orderable quandles are given.
Paper studies geometric and combinatorial properties of circular snakes.
problem Exploring geometric and combinatorial properties of circular snakes.
method Definition and investigation of outer Lipschitz geometry, decomposition of Valette link, construction of combinatorial objects, weakly outer Lipschitz classification.
result Existence of canonical decomposition and necessary/sufficient criteria for removing segments or Hölder triangles.
The study identifies surfaces with Maslovian normal bundles.
problem Characterizing surfaces with specific geometric properties.
method Proving equivalence to round spheres, cylinders, or cones.
result Surfaces with Maslovian normal bundles are limited to specific shapes.
New hexagonal circular 3-webs with reducible curves classified.
problem Classifying hexagonal circular 3-webs with reducible polar curves of degree 3.
method New examples and classifications presented.
result Classification of hexagonal circular 3-webs with reducible polar curves of degree 3.
Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.
problem Discretizing surfaces with spherical curvature lines.
method Lie-geometric discretisation in terms of principal contact element nets.
result Circular nets with two families of spherical parameter lines are related to Darboux cyclides.
Paper proposes a method to improve circular coordinate representation for detecting changes in high-dimensional datasets.
problem Detecting changes in high-dimensional datasets with preserved topological structures.
method Adapt circular coordinate framework using a generalized penalty function instead of an L2 penalty.
result Circular coordinates with generalized penalty can detect changes in high-dimensional datasets under different sampling schemes.
The paper studies circular evolutes and involutes of framed curves in Euclidean space.
problem Investigating properties of framed curves and their evolutes and involutes.
method Definition and analysis of circular evolutes and involutes of framed curves, properties of normal surfaces, and their relations.
result Circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similar to fronts in the Euclidean plane.
CDFD analyzes circularity and directionality in weighted directed networks.
problem Analyzing circularity and directionality in weighted directed networks.
method CDFD framework separates flow into circular and acyclic components.
result CDFD yields a normalized circularity index capturing flow in cycles and directionality.
It was shown by Kaup that every origin-preserving automorphism of quasi-circular domains is a polynomial mapping. In this paper, we study how the weight of quasi-circular domains and the degree of such automorphisms are related. By using the Bergman mapping, we prove that every origin-preserving automorphism of normal …
Classifies hexagonal circular 3-webs with cubic polar curves.
problem Classifying hexagonal circular 3-webs with algebraic polar curves of degree three.
method Analyzes hexagonal circular 3-webs on unit sphere with polar points on a twisted cubic.
result Completes the classification of hexagonal circular 3-webs with algebraic polar curves of degree three.
Study geometrically characterizes piecewise circular curves with decreasing curvature.
problem Characterizing piecewise circular curves with decreasing curvature.
method Introducing moduli spaces and relating them to Legendrian polygons.
result Proves the moduli space contains a connected component homeomorphic to the Fock-Goncharov space of positive flags.
Circular disc can be tiled with up to 3 congruent pieces, showing symmetry.
problem Tiling a circular disc with congruent pieces.
method Proving the existence of a k-fold rotational symmetry for k≤3. result First nontrivial estimate on minimum number of tiles for certain tiling configurations.
The paper extends circular variable modelling tools from statistics to machine learning.
problem Circular variables in various fields are poorly modeled by current tools.
method Introduces mGvM distribution and circular regression model.
result Efficient inference and learning methods for circular data.
A new distance measure for circular Heegaard splittings helps understand knot exteriors.
problem Understanding the structure of knot exteriors using circular Heegaard splittings.
method Defining and analyzing circular distance for circular Heegaard splittings.
result Circular distance bounds properties of knot exteriors, like the uniqueness of minimal-genus Seifert surfaces.
Improved curvature estimation for digital objects.
problem Estimating curvature profiles in digital images.
method Extending MDCA curvature estimator to λ-MDCA, proving convergence, and evaluating performance.
result λ-MDCA curvature estimator converges with order $\mathcal{O}(h^{ rac13})$ and performs better near corners.
Curves become nearly circular over time without initial assumptions.
problem Understanding the asymptotic behavior of area-preserving flows.
method Proving asymptotic circularity without additional assumptions.
result Immortal solutions become asymptotically circular without initial assumptions.
2x2 Lax representation for circular nets with constant negative Gauss curvature.
problem Discrete circular nets with constant negative Gauss curvature.
method 2x2 Lax representation, associated family, and Bäcklund transformations.
result All members of the associated family have constant Gauss curvature.
Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.
problem Understanding free path lengths on flat surfaces with circular obstacles.
method Proved the existence of a limiting distribution using radius of obstacles as a parameter.
result Relates the limiting distribution to heights of zippered rectangle decompositions.
The paper classifies and studies symplectic and contact properties of circular spherical divisors.
problem Investigating symplectic and contact topology of circular spherical divisors.
method Classification and analysis of concave circular spherical divisors, including embedding, Stein fillability, and rational homology type determination.
result All concave circular spherical divisors up to toric equivalence are realized as symplectic log Calabi-Yau pairs with minimal complements.
We show that circular width is preserved under connected sum of knots for some cases.
In this paper, we show the existence of real-analytic stationary Navier-Stokes flows with isotropic streamlines in all latitudes in some simply-connected flow region on a rotating round sphere. We also exclude the possibility of having a Poiseuille's flow profile to be one of these stationary Navier-Stokes flows with i…
The paper explores circular orderability in 3-manifold groups, related to the L-space conjecture.
problem Circular orderability of 3-manifold groups and its relation to the L-space conjecture.
method Investigation of finite cyclic covers and Dehn surgeries to establish circular orderability.
result Circularly orderable fundamental groups of compact, connected, P^2-irreducible 3-manifolds are characterized.
Minimal triangulations of circle bundles linked to circular permutations.
problem Which circle bundles can be triangulated over a given base triangulation?
method Minimal triangulations encoded by local systems of circular permutations of vertices.
result Classical Huntington transitivity axiom for cyclic orders expressed as a binary Chern cocycle.
ANGLE tackles circular data regression, improving predictive performance.
problem Geometrically misleading traditional regression for circular data.
method Generative map optimized via GCES loss for non-parametric distributional regression.
result Unified toolbox for circular statistics challenges.
Derives conformal parameters of curves using inscribed circular polygons.
problem Characterizing conformal invariants of smooth curves in 3D.
method Limiting process with inscribed circular polygons, based on elementary geometry.
result Derives conformal length, curvature, and torsion via a novel method.
New methods map circular discs to squares smoothly and invertibly.
problem Mapping circular discs to squares while maintaining desirable properties.
method Analytical expressions for smooth, invertible mappings with specific properties.
result Presented new mappings with conformal, equiareal, and radially-constrained properties.
Machine learning predicts wind pressures around circular cylinders efficiently.
problem Predicting wind pressures around circular cylinders using traditional methods is costly and time-consuming.
method Trained GBRT models using Reynolds number, turbulence intensity, and circumferential angle as inputs.
result GBRT models accurately predict wind pressures for a wide range of Reynolds and turbulence intensities.
New moves transform any virtual knot to a trivial knot.
problem Transforming virtual knots to trivial knots.
method Introducing arc shift and region arc shift moves.
result Any virtual knot can be transformed into a trivial knot using these moves.
Defines knotting probability for spatial arcs.
problem Calculating the probability of knotting in spatial arcs.
method Projection of spatial arcs to planes, defining knotting probability for diagrams.
result Knotting probability defined for every oriented spatial arc.
Study circular orders on groups, showing recursive enumerability and describing actions.
problem Understanding circular orders on groups and their properties.
method Topology on CO(G), compactness, action of Aut(G), classification of elements.
result Recursively enumerable sets of non-linearly and non-circularly orderable groups.
A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\N (K), with the property that every regular l…
This article provides sufficient conditions for a closed hyperbolic 3-manifold M with non zero first Betti number to fiber over the circle, and to find a fiber in M. Those conditions are formulated in terms of the behavior the circular characteristic in finite regular covers of M. We define the circular character…
A cone projection maps complex plane arcs to conic sections with fixed focus and directrix.
problem Mapping complex plane arcs to conic sections with fixed focus and directrix.
method Elementary spatial construction and reciprocal lens identity.
result The cone projection maps every line not through the origin onto an arc of a conic with focus at the origin and directrix the line itself.
Study on unknotting twisted knots using arc shift and region arc shift moves.
problem Unknotting twisted knots and finding bounds for region arc shift number.
method Introduced arc shift move and region arc shift move for twisted knots.
result Found families of twisted knots with specific arc shift and region arc shift numbers.